A critique of impedance measurements in cardiac tissue

The specific impedance of cardiac tissue cannot be measured directly. Instead, the investigator obtains voltage and current measurements and places them into a model of the tissue's structure to infer the impedances of elements of the model. If the model fails to describe major aspects of the real tissue, the results may be worthless, although possibly self-consistent. In the literature of impedance measurement in cardiac tissue, only rarely is the model explicitly described; more commonly, the tissue model is adopted implicitly when equations giving the impedance in terms of voltage and current measurements are adopted. This paper examines the series of models that have been used in specific impedance measurements of cardiac tissue and shows how the same or similar measurements can accurately describe tissue impedivity or can lead to significant errors when inadequate models such as isotropic and anisotropic monodomains (although a part of work of historical merit) are used.     

A planar slab bidomain model for cardiac tissue

A fully three-dimensional model of the ventricular or atrial free wall will involve a planar geometry of finite thickness. The governing equations for the interstitial and extracellular potential of a planar slab of cardiac tissue comprised of parallel fibers undergoing uniform plane-wave activation are presented. A comparison with a bidomain of cylindrical geometry with the same half-thickness shows that the potentials in the planar bidomain (as a function of depth) approach core-conductor behavior more quickly.     

Three-dimensional pseudospectral modelling of cardiac propagation in an inhomogeneous anisotropic tissue

Various investigators have used the monodomain model to study cardiac propagation behaviour. In many cases, the governing non-linear parabolic equation is solved using the finite-difference method. An adequate discretisation of cardiac tissue with realistic dimensions, however, often leads to a large model size that is computationally demanding. Recently, it has been demonstrated, for a two-dimensional homogeneous monodomain, that the Chebyshev pseudospectral method can offer higher computational efficiency than the finite-difference technique. Here, an extension of the pseudospectral approach to a three-dimensional inhomogeneous case with fibre rotation is presented. The unknown transmembrane potential is expanded in terms of Chebyshev polynomial trial functions, and the monodomain equation is enforced at the Gauss-Lobatto node points. The forward Euler technique is used to advance the solution in time. Numerical results are presented that demonstrate that the Chebyshev pseudospectral method offered an even larger improvement in computational performance over the finite-difference method in the three-dimensional case. Specifically, the pseudospectral method allowed the number of nodes to be reduced by ≈85 times, while the same solution accuracy was maintained. Depending on the model size, simulations were performed with ≈18–41 times less memory and ≈99–169 times less CPU time.     

Mechanism for polarisation of cardiac tissue at a sealed boundary

If current is flowing in cardiac tissue, and if the myocardial fibres approach a sealed boundary at an angle, then the tissue within a few length constants of the boundary is polarised. This polarisation occurs when the cardiac tissue has different anisotropy ratios in the intracellular and extracellular spaces. This new mechanism of tissue polarisation is demonstrated using a simple, analytical model, and it is shown quantitatively that this polarisation can be nearly as large as that occurring near an electrode.     

Anisotropic Mechanisms for Multiphasic Unipolar Electrograms: Simulation Studies and Experimental Recordings

The origin of the multiple, complex morphologies observed in unipolar epicardial electrograms, and their relationships with myocardial architecture, have not been fully elucidated. To clarify this problem we simulated electrograms (EGs) with a model representing the heart as an anisotropic bidomain with unequal anisotropy ratio, ellipsoidal ventricular geometry, transmural fiber rotation, epi-endocardial obliqueness of fiber direction and a simplified Purkinje network. The EGs were compared with those directly recorded from isolated dog hearts immersed in a conducting medium during ventricular excitation initiated by epicardial stimulation. The simulated EGs share the same multiphasic character of the recorded EGs. The origin of the multiple waves, especially those appearing in the EGs for sites reached by excitation wave fronts spreading across fibers, can be better understood after splitting the current sources, the potential distributions and the EGs into an axial and a conormal component and after taking also into account the effect of the reference or drift component. The split model provides an explanation of humps and spikes that appear in the QRS (the initial part of the ventricular EG) wave forms, in terms of the interaction between the geometry and direction of propagation of the wave front and the architecture of the fibers through which excitation is spreading. © 2000 Biomedical Engineering Society. PAC00: 8719Nn, 8710+e, 8719Hh     

A Finite Volume Method for Modeling Discontinuous Electrical Activation in Cardiac Tissue

This paper describes a finite volume method for modeling electrical activation in a sample of cardiac tissue using the bidomain equations. Microstructural features to the level of cleavage planes between sheets of myocardial fibers in the tissue are explicitly represented. The key features of this implementation compared to previous modeling are that it represents physical discontinuities without the implicit removal of intracellular volume and it generates linear systems of equations that are computationally efficient to construct and solve. Results obtained using this method highlight how the understanding of discontinuous activation in cardiac tissue can form a basis for better understanding defibrillation processes and experimental recordings.     

Two-dimensional Chebyshev pseudospectral modelling of cardiac propagation

Bidomain or monodomain modelling has been used widely to study various issues related to action potential propagation in cardiac tissue. In most of these previous studies, the finite difference method is used to solve the partial differential equations associated with the model. Though the finite difference approach has provided useful insight in many cases, adequate discretisation of cardiac tissue with realistic dimensions often requires a large number of nodes, making the numerical solution process difficult or impossible with available computer resources. Here, a Chebyshev pseudospectral method is presented that allows a significant reduction in the number of nodes required for a given solution accuracy. The new method is used to solve the governing nonlinear partial differential equation for the monodomain model representing a two-dimensional homogeneous sheet of cardiac tissue. The unknown transmembrane potential is expanded in terms of Chebyshev polynomial trial functions and the equation is enforced at the Gauss-Lobatto grid points. Spatial derivatives are obtained using the fast Fourier transform and the solution is advanced in time using an explicit technique. Numerical results indicate that the pseudospectral approach allows the number of nodes to be reduced by a factor of sixteen, while still maintaining the same error performance. This makes it possible to perform simulations with the same accuracy using about twelve times less CPU time and memory.     

A Deformable Finite Element Derived Finite Difference Method for Cardiac Activation Problems

We present a finite element (FE) derived finite difference (FD) technique for solving cardiac activation problems over deforming geometries using a bidomain framework. The geometry of the solution domain is defined by a FE mesh and over these FEs a high resolution FD mesh is generated. The difference points are located at regular intervals in the normalized material space within each of the FEs. The bidomain equations are then transformed to the embedded FD mesh which provides a solution space that is both regular and orthogonal. The solution points move in physical space with any deformation of the solution domain, but the equations are set up in such a way that the solution is invariant as it is constructed in material space. The derivation of this new solution technique is presented along with a series of examples that demonstrate the accuracy of this bidomain framework. © 2003 Biomedical Engineering Society. PAC2003: 8719Hh, 8710+e, 8719Rr     

Dependence of cardiac strength-interval curves on pacing rate

The purpose of the research is to determine how the pacing rate affects the strength-interval curve in cardiac tissue. Computer simulations are used to calculate the cathodal and anodal strength-interval curves. The tissue is represented by the bidomain model with Beeler-Reuter membrane properties. The strength-interval curves shift to shorter intervals as the pacing rate increases. However, the shape of the strength-interval curve, including the separation into ‘make’ and ‘break’ sections and the presence of a ‘dip’, is insensitive to pacing rate.